Miskolc Mathematical Notes Vol. 16 (2015), No. 2, pp. 869–885

HU e-ISSN 1787-2413 DOI: 10.18514/MMN.2015.1308

PARAMETRIC MARCINKIEWICZ INTEGRALS ON WEIGHTED HERZ SPACES YUE HU AND YUESHAN WANG Received 05 September, 2014 

Abstract. Let 0 <  < n and ˝ be the parametric Marcinkiewicz integral. In this paper we shall  ˛;p obtain the strong type and weak type estimates of ˝ on the weighted Herz spaces KP q .!1 ; !2 / with general weights. The boundedness of the commutators generated by BMO functions and parametric Marcinkiewicz integral is also obtained. 2010 Mathematics Subject Classification: 42B25; 42B35 Keywords: parametric Marcinkiewicz integral, commutator, Muckenhoupt weight, BMO, Herz spaces, weak Herz space

1. I NTRODUCTION AND RESULTS Suppose that Sn 1 is the unit sphere in Rn .n  2/ equipped with the normalized Lebesgue measure d: Let ˝ be a homogeneous function of degree zero on Rn satisfying ˝ 2 L1 .Sn 1 / and Z ˝.x 0 /d .x 0 / D 0; (1.1) Sn

1

where x 0 D x=jxj for any x ¤ 0: For 0 <  < n; H¨ormander in [6] defined the para metric Marcinkiewicz integral operator ˝ of higher dimension as follows.  ˝ .f

Z /.x/ D

0

1

dt  jF˝;t .x/j2 2C1 t

1=2 ;

where  F˝;t .x/

Z D

jx yjt

˝.x y/ f .y/dy: jx yjn 

The first author was supported in part by the National Natural Science Foundation of China, Grant No.11071057. c 2015 Miskolc University Press

870

YUE HU AND YUESHAN WANG

Let b be a locally integrable function, the commutator generated by parametric Mar cinkiewcz integral ˝ and b is defined by ! 12 ˇ2 Z 1 ˇZ ˇ ˇ dt ˝.x y/  ˇ Œb; ˝ .f /.x/ D .b.x/ b.y//f .y/dy ˇˇ 2C1 : ˇ n  t 0 jx yjt jx yj When  D 1; we shall denote 1˝ simply by ˝ : The area of the Marcinkiewcz integrals have been under intensive research. This operator ˝ was first introduced by Stein in [15]. He proved that if ˝ 2 Lip˛ .Sn 1 /.0 < ˛  1/; then ˝ is the operator of strong type .p; p/ for 1 < p  2 and of weak type .1; 1/: Here, we say that ˝ 2 Lip˛ .Sn 1 / if j˝.x 0 /

˝.y 0 /j  jx 0

y 0 j˛ ; x 0 ; y 0 2 Sn

1

:

In 1990, Torchinsky and Wang in [16] considered the weighted case and proved that If ˝ 2 Lip˛ .Sn 1 /.0 < ˛  1/; and b 2 BMO.Rn /; then for all 1 < p < 1; and ! 2 Ap (Muckenhoupt weight class), ˝ and Œb; ˝  are all bounded in Lp .!/: On the other hand, in 1960, H¨ormander [6] showed that if ˝ 2 Lip˛ .Sn 1 /.0 < ˛  1/;  then for 0 <  < n; ˝ is of strong type .p; p/ for all 1 < p < 1: In [14] Shi and Jiang obtained the weighted Lp -boundedness of parametric Marcinkiewicz integral and its commutator. Theorem A ([14]). Let 0 <  < n and ˝ 2 L1 .Sn 1 /: If ! 2 Ap .1 < p < 1/ and b 2 BMO.Rn /; then there exists a constant C > 0 independent of f such that 

k˝ .f /kLp .!/  C kf kLp .!/ and 

kŒb; ˝ .f /kLp .!/  C kbk kf kLp .!/ : Notice that Lip˛ .Sn 1 /.0 < ˛  1/ ¤ L1 .Sn 1 / ¤ L1 .Sn 1 /; then the results of Theorem A are extend and improve the results of Torchinsky and Wang in [16]. Let us recall the definition of Littlewood-Paley function Z 1=2 2 dt S˚ .f /.x/ D j˚ t  f .x/j ; t Rn R where ˚ t .x/ D t n ˚.x=t / and ˚ 2 L1 .Rn / which satisfies Rn ˚.x/dx D 0: It is well known that the Littlewood-Paley function have long played important roles in harmonic analysis. When ˚ is given by ˚.x/ D jxj

nC

˝.x/Œ0;1 .jxj/;

where ˝ be a homogeneous function of degree zero on Rn satisfying (1.1), then ˚.x/ becomes the parametric Marcinkiewicz integral operator. Therefore, many au thors have been interested in studying the boundedness properties of ˝ on various function spaces, it can be seen in [1–3, 18].

PARAMETRIC MARCINKIEWICZ INTEGRALS ON WEIGHTED HERZ SPACES

871

In [13], Lu, Yabuta and Yang obtained the boundedness results for sublinear operators on weighted Herz spaces with general Muckenhoupt weights. Recently, many authors considered the boundedness of operators on weighted Herz spaces with general Muckenhoupt weights. In [8], Komori and Matsuoka showed the boundedness of singular integral operators and fractional integrals on weighted Herz spaces. In [5], Guo and Jiang discussed the boundedness of commutators of singular integral operators on weighted Herz spaces, and as an application, they obtained the interior estimates on weighted Herz spaces for the solutions of some nondivergence elliptic equations. Hu , He and Wang [19] studied the boundedness of commutators of fractional integrals in generalized Herz spaces. More results concerning the boundedness of operators on Herz spaces can be seen in [12, 17]. The main purpose of this paper is to consider the boundedness of parametric Marcinkiewicz integral on weighted Herz spaces with Ap weights. At the extreme case,  we will also prove that ˝ is bounded from the weighted Herz spaces to the weighted weak Herz spaces. The boundedness of commutators generated by parametric Marcinkiewicz integral and BMO functions on weighted Herz spaces is also considered. Our main results in the paper are formulated as follows. Theorem 1. Let 0 <  < n and ˝ 2 L1 .Sn 1 /: Suppose 0 < p < 1; 1 < q <  ˛;p 1; !1 2 Aq1 and !2 2 Aq2 : Then ˝ is bounded on KP q .!1 ; !2 / provided that !1 and !2 satisfy either of the following (i) !1 D !2 ; 1  q1 D q2  q and nq1 =q < ˛q1 < n.1 q1 =q/I (ii) !1 ¤ !2 ; 1  q1 < 1; 1  q2  q and 0 < ˛q1 < n.1 q2 =q/: Theorem 2. Let 0 <  < n and ˝ 2 L1 .Sn 1 /: Suppose 0 < p < 1; 1 < q < 1; !1 2 Aq1 and !2 2 Aq2 : If 1  q1 < 1; 1  q2  q and ˛q1 D n.1 q2 =q/, then  ˛;p ˛;p ˝ is bounded from KP q .!1 ; !2 / to W KP q .!1 ; !2 /: Theorem 3. Let 0 <  < n and ˝ 2 L1 .Sn 1 /: Suppose 1 < q < 1; !1 2 Aq1 ;  ˛;p !2 2 Aq2 and b 2 BMO: Then Œb; ˝  is bounded on KP q .!1 ; !2 / provided that !1 and !2 satisfy either of the following (i) !1 D !2 ; 1  q1 D q2  q and nq1 =q < ˛q1 < n.1 q1 =q/I (ii) !1 ¤ !2 ; 1  q1 < 1; 1  q2  q and 0 < ˛q1 < n.1 q2 =q/: Throughout this paper, unless otherwise indicated, C will be used to denote a positive constant that is not necessarily the same at each occurrence. 2. D EFINITIONS AND PRELIMINARIES We begin this section with some properties of Ap weights which play important role in the proofs of our main results. A weight ! is a nonnegative, locally integrable function on Rn : Let B D B.x0 ; r/ denote the ball with the center x0 and radius r, and let B D B.x0 ; r/ for any  > 0:

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YUE HU AND YUESHAN WANG

For a given weight function ! and a measurable set E, we R also denote the Lebesgue measure of E by jEj and set weighted measure !.E/ D E !.x/dx: Definition 1. A weight ! is said to belong to Ap for 1 < p < 1; if there exists a constant C such that for every ball B  Rn ;   p 1 Z Z 1 1 1 p0 !.x/dx !.x/ dx  C; (2.1) jBj B jBj B where s 0 is the dual of s such that 1=s C 1=s 0 D 1: The class A1 is defined by replacing the above inequality with Z 1 !.y/dy  C
ess inf w.x/: jBj B x2B By (2.1), we have Z

 Z !.x/dx

B

!.x/1

p0

p

1

dx

B

 C jBjp :

(2.2)

The classical Ap weight theory was first introduced by Muckenhoupt in the study of weighted Lp -boundedness of Hardy-Littlewood maximal function in [10]. Lemma 1 ([4, 10]). Let 1  p < 1 and ! 2 Ap : Then the following statements are true: (i) There exists constant C such that !.Bk /  C 2np.k !.Bj /

j/

for k > j I

(2.3)

(ii) For any 1 < p < 1; there exists 1 < q < p such that ! 2 Aq .Rn /I (iii) There exist two constants C and ı > 0 such that for any measurable set E  B;   !.E/ jEj ı C I (2.4) !.B/ jBj (iv) For any 1 < p < 1; one has ! 1

p0

2 Ap0 .Rn /:

If ! satisfies (2.4), we say ! 2 A1 .Rn /: It is well known that [ A1 .Rn / D Ap .Rn /: 1p 0 and any measurable function f on Rn ; we write Ek .; f / D fx 2 Ck W jf .x/j > g: Definition 3 ([13]). Let ˛ 2 R; 0 < p; q < 1 and !1 ; !2 be two weight function on Rn : A measurable function f on Rn is said to belong to the homogeneous weighted ˛;p weak Herz space W KP q .!1 ; !2 / if !1=p 1 X !1 .Bk /˛p=n Œ!2 .Ek .; f //p=q < 1: kf kW KP q˛;p .!1 ;!2 / D sup  >0

kD 1

0;p 0;p Obviously, if ˛ D 0; then KPp .!1 ; !2 / D Lp .!2 / and W KPp .!1 ; !2 / p D W L .!2 / for any 0 < p < 1: Thus, weighted Herz spaces are generalizations of the weighted Lebesgue spaces.

Definition 4. A locally integrable function b is said to be in BMO.Rn / if Z 1 sup jb.x/ bB jdx D kbk < 1; B jBj B R 1 where bB D jBj B b.y/dy: Lemma 2. (John-Nirenberg inequality, see [7]) Let b 2 BMO.Rn /: Then for any ball B  Rn ; there exist constant C1 ; C2 such that for all  > 0; jfx 2 B W jb.x/

bB j > gj  C1 jBjexp. C2 =kbk /:

Lemma 3 ([11]). Let ! 2 A1 : Then the norm of BMO.!; Rn / is equivalent to the norm of BMO.Rn /; where Z ˚ 1 n jb.x/ bB;! j!.x/dx ; BMO.!; R / D b W kbk;! D sup BRn !.B/ B and bB;!

1 D !.B/

Z b.´/!.´/d´: B

Lemma 4. Suppose ! 2 A1 .Rn /; b 2 BMO.Rn /: Then for any p  1 we have  1=p Z 1 jb.x/ bB;! jp !.x/dx  C kbk : !.B/ B

874

YUE HU AND YUESHAN WANG

Proof. Since !.x/ 2 A1 .Rn /; then (iii) of Lemma 1 and Lemma 2 imply ! .fx 2 B W jb.x/

bB j > g/  C !.B/exp. C2 ı=kbk /:

So Z B

jb.x/

1

Z

p

pp 1 ! .fx 2 B W jb.x/ bB j > g/ d  0 Z 1  C !.B/ pp 1 exp. C2 ıkbk /d 

bB j !.x/dx 

0

p

 C !.B/kbk : Thus 1 !.B/

C bB;! j !.x/dx  !.B/

Z B

p

jb.x/

Z B

jb.x/

p

bB jp !.x/dx  C kbk : 

3. P ROOF OF T HEOREM 1 Let f 2

˛;p KP q .!1 ; !2 /: Then



 .f / p ˛;p ˝ KP q .!1 ;!2 /

0

1 X

C

!1 .Bj /


˛p n

@

jD 1 1 X

CC

jX 2 kD 1

0 !1 .Bj /


˛p n

CC

jX C1

@

jD 1 1 X

1p



. .f k /j .x/ q A ˝ L .!2 / 1p



. .f k /j .x/ q A ˝ L .!2 /

kDj 1

1p 1 X



. .f k /j .x/ q @ A ˝ L .!2 / 0

!1 .Bj /


˛p n

jD 1

kDj C2

D E1 C E2 C E3 : 

By the fact that ˝ is a bounded operator on Lq .!2 /, we get 0 jX C1 1 X




˛p

. .f k /j .x/ E2 DC !1 .Bj / n @ ˝

jD 1

C 

1 X

kDj 1

1p jX C1


˛p

f k q A !1 .Bj / n @ L .!2 /

jD 1 p C kf k P ˛;p : Kq .!1 ;!2 /

0

kDj 1

Lq .!2 /

1p A

PARAMETRIC MARCINKIEWICZ INTEGRALS ON WEIGHTED HERZ SPACES

875

For any x 2 Cj and y 2 Ck \fy W jx yj < tg with j  k C2, we have t > jx yj  jxj jyj  jxj=2  C 2j n : So !1=2 ˇ2 Z 1 ˇZ ˇ ˇ ˇ  ˇ dt ˝.x y/ ˇ .f k /j .x/ˇ D ˇ f .y/dy ˇˇ 2C1 ˝ ˇ n  k t 0 jx yjt jx yj 1=2 Z 1 dt j.n /  C2 kf k kL1 : 2C1 C 2j n t Then ˇ  ˇ ˇ .f k /j .x/ˇ D C 2 ˝

jn

kf k kL1 ;

(3.1)

and



˝ .f k /j .x/

Lq .!2 /

jn

 C2

!2 .Bj /1=q kf k kL1 :

By H¨older’s inequality, kf k kL1

Z  C kf k kLq .!2 / .

q0

!2 .x/1

0

dx/1=q :

(3.2)

Bk

Since !2 2 Aq2 .Rn /  Aq .Rn /; by (2.2) and (2.3) we get, Z Z
1=q 1 q0 1=q 0 !2 .x/ . dx/ !2 .x/dx Bk

Bj

Z D.

1 q0

1=q 0

!2 .x/ dx/ Bk k nC.j k/nq2 =q

 C2

Z !2 .x/dx


1=q



Bk

!2 .Bj / !2 .Bk /

1=q

:

Then, for j  k C 2 we have



˝ .f k /j .x/

Lq .!2 /

(3.3)

 C kf k kLq .!2 / 2.j

k/n.q2 =q 1/

:

Thus E1 D C

0 1p jX 2




˛p

 .f k /j .x/ q A !1 .Bj / n @ ˝ L .!2 /

1 X jD 1 p

 C kbk

kD 1 1 X

0 @

jD 1 p

 C kbk

jX 2

1 X


!1 .Bj / kf k kLq .!2 / 2.j

k/n.q2 =q 1/ A

kD 1

0

jX 2

@ jD 1

1p ˛ n

kD 1

1p ˛

.!1 .Bk // n kf k kLq .!2 / 2.j

k/.˛q1 Cq2 n=q n/ A

:

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YUE HU AND YUESHAN WANG

When 0 < p  1; we get E1 

p

 C kbk 

1 X

p C kbk

.!1 .Bk //

kD 1 1 X

˛p n

kC2 X

p kf k kLq .!2 /

2.j

k/p.˛q1 Cq2 n=q n/

jD 1

.!1 .Bk //

˛p n

p

kf k kLq .!2 /

kD 1 p p C kbk kf k P ˛;p : Kq .!1 ;!2 /.Rn /

When p > 1; by H¨older’s inequality we get 0 jX 2 1 X ˛p p p @ .!1 .Bk // n kf k kLq .!2 / 2.j E1  C kbk jD 1

0

1 k/.˛q1 Cq2 n=q n/ A

kD 1

1p=p0

jX 2

2.j

@

k/.˛q1 Cq2 n=q n/ A

kD 1 p

 C kbk

p

1 X

.!1 .Bk //

˛p n

p

kf k kLq .!2 /

kD 1 p : ˛;p KP q .!1 ;!2 /.Rn /

 C kbk kf k

Let us now turn to estimate the last term E3 . In the case k  j C 2, for any x 2 Cj and y 2 Ck \ fy W jx yj < tg, we have t > jx yj  jyj jxj  jyj=2  C 2k n : Then, similar to the estimates of (3.1), ˇ  ˇ ˇ .f k /j .x/ˇ  C 2 k n kf k kL1 : (3.4) ˝ So,



˝ .f k /j .x/

 C2

Lq .!2 /

By (2.2) and (2.4), Z . !2 .x/1 Bk

q0

1=q 0

kn

Z

dx/

!2 .x/dx Bj

!2 .Bj /1=q kf k kL1 :


1=q

 C 2k nC.j

Combining with (3.2), (3.5) and (3.6), we have



 C kf k kLq .!2 / 2.j

˝ .f k /j .x/ q L .!2 /

for j  k

2:

k/ı2 n=q

k/ı2 n=q

(3.5)

:

(3.6)

PARAMETRIC MARCINKIEWICZ INTEGRALS ON WEIGHTED HERZ SPACES

877

When 0 < p  1; we get E3 

1 X

p C kbk

!1 .Bj /

jD 1



1 X

p C kbk

1 X

˛p n

p

kf k kLq .!2 / 2.j

kDj C2

.!1 .Bk //

˛p n

p kf k kLq .!2 /

 

p C kbk

k X2



jD 1

kD 1 1 X

k/pı2 n=q

.!1 .Bk //

˛p n

p kf k kLq .!2 /

k X2

!1 .Bj / !1 .Bk /

2.j

p

 C kbk

1p k/.ı1 ˛Cı2 n=q/ A

kDj C2

0

1 X

@ jD 1

0

k/pı2 n=q

k/p.ı1 ˛Cı2 n=q/

When q > 1; by H¨older’s inequality we get 0 1 1 X X ˛ p @ E3  C kbk .!1 .Bk // n kf k kLq .!2 / 2.j 1 X

2.j

jD 1

kD 1 p p C kbk kf k P ˛;p : Kq .!1 ;!2 /

jD 1

 ˛p n

1 .!1 .Bk //

˛p n

p

kf k kLq .!2 / 2.j

k/.ı1 ˛Cı2 n=q/ A

kDj C2

1p=p0

1 X

2.j

@

k/.ı1 ˛Cı2 n=q/ A

kDj C2

 

p C kbk

1 X

.!1 .Bk //

˛p n

k X2

p kf k kLq .!2 /

2.j

k/.ı1 ˛Cı2 n=q/

jD 1

kD 1 p p C kbk kf k P ˛;p : Kq .!1 ;!2 /

Combining the above estimates for E1 ; E2 and E3 , the proof of Theorem 1 is completed. 4. P ROOF OF T HEOREM 2 ˛;p

Let f 2 KP q .!1 ; !2 /: Then 1 X

p


˛p
p=q  !1 .Bj / n !2 .fx 2 Cj W j˝ .f /.x/j > g/

jD 1

 p

1 X jD 1

0
!1 .Bj /

˛p n

@!2

n

x 2 Cj W

jX 2 kD 1

1p=q 

j˝ .fk /.x/j > =3

o A

878

YUE HU AND YUESHAN WANG 1 X

C p

0
!1 .Bj /

˛p n

@!2

n

x 2 Cj W

1p=q

jX C1



j˝ .fk /.x/j > =3

o A

jD 1

kDj 1

1 X

1p=q 1 n o X
˛p  j˝ .fk /.x/j > =3 A !1 .Bj / n @!2 x 2 Cj W

jD 1

kDj 2

C p

0

D F1 C F2 C F3 : Applying Chebyshev’s inequality [4] and Theorem A, we obtain 1 X

F2  C p

0 !1 .Bj /




˛p n

jD 1

C

1 X

@ 1 q



q

k˝ .fk /kLq .!2 / A

kDj 1

1p jX C1


˛p

f k q A !1 .Bj / n @ L .!2 / 0

jD 1 p C kf k P ˛;p : Kq .!1 ;!2 /



1p=q

jX C1

kDj 1

For any x 2 Cj and y 2 Ck \ fy W jx (3.1), (3.2) and (3.3) we have ˇ ˇ
ˇ ˇ  ˇ˝ .f k /j .x/ˇ  C !2 .Bj /

yj < t g with j  k C 2, by the inequalities

1=q

j nCk nC.j k/nq2 =q

2

Noting the fact ˛q1 D n.1 q2 =q/; then ˇ ˇ
ˇ ˇ  ˇ˝ .f k /j .x/ˇ  C !2 .Bj /

1=q .k j /˛q1

2

kf k kLq .!2 / :

kf k kLq .!2 / :

Moreover, since 0 < p  1; then for any x 2 Cj ; jX 2

ˇ  ˇ ˇ .fk /.x/ˇ ˝

kD 1


 C !2 .Bj / jX 2

1=q

.k j /˛q1

2

˛=n


!1 .Bj /



˛ n

.!1 .Bk // kf k kLq .!2 /

kD 1


 C !2 .Bj /

1=q


!1 .Bj /

˛=n

jX 2 kD 1

!1 .Bj / !1 .Bk / ˛

 ˛n

.!1 .Bk // n kf k kLq .!2 /

PARAMETRIC MARCINKIEWICZ INTEGRALS ON WEIGHTED HERZ SPACES

0
 C !2 .Bj /

1=q


!1 .Bj /


 C !2 .Bj /

1=q


!1 .Bj /

jX 2

˛=n @

11=p .!1 .Bk //

˛p n

p

kf k kLq .!2 / A

kD 1 ˛=n

kf kKP q˛;p .!1 ;!2 / :

If n

x 2 Cj W

jX 2

o  j˝ .fk /.x/j > =3 D ¿;

kD 1

then p ˛;p KPp .!1 ;!2 /

F1  C kf k holds is trivially. Now we suppose n

x 2 Cj W

jX 2

o  j˝ .fk /.x/j > =3 ¤ ¿:

kD 1

Let
Sj D !2 .Bj /

1=q


!1 .Bj /

˛=n

:

Since ˛ > 0; it is easy to see that lim Sj D 0:

j !1

Then for any  > 0; we can find a maximal positive integer j such that =3  CSj kf kKP q˛;p .!1 ;!2 / : So p

F1  

j X

!2 .Bj /


p=q


˛p=n !1 .Bj /

jD 1

 C kf

p k P ˛;p Kp .!1 ;!2 /

 C kf k  C kf

p ˛;p KPp .!1 ;!2 /

 ˛p  p j  X !1 .Bj / n !2 .Bj / q !1 .Bj / !2 .Bj /

jD 1 j X

2.j

j /.ı1 ˛pCı2 pn=q/

jD 1 p k P ˛;p : Kq .!1 ;!2 /

Let us now estimate F3 . From (3.2), (3.4) and (3.6) we have ˇ ˇ
1=q .j k/ı n=q ˇ  ˇ 2 2 kf k kLq .!2 / ˇ˝ .f k /j .x/ˇ  C !2 .Bj /

879

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YUE HU AND YUESHAN WANG

for j  k

2: So

1 X



j˝ .fk /.x/j

kDj 2 1=q


 C !2 .Bj / 1 X


!1 .Bj /

˛=n

.!1 .Bk //

˛=n

.j k/ı2 n=q

kf k kLq .!2 / 2

kDj 2


 C !2 .Bj /
 C !2 .Bj /

1=q

1=q


!1 .Bj /


!1 .Bj /

˛=n

1 X

˛=n

kDj 2 1 X



!1 .Bj / !1 .Bk /

˛=n

.!1 .Bk //˛=n kf k kLq 2.j

k/.ı2 n=qC˛ı1 /

.!1 .Bk //˛=n kf k kLq :

kDj 2

Noting that 0 < p  1; we have 1 X



j˝ .fk /.x/j

kDj 2

0
 C !2 .Bj /

1=q


!1 .Bj /


 C !2 .Bj /

1=q


!1 .Bj /

˛=n @

1 X

11=p p

j .!1 .Bk //˛p=n kf k kLq A

kDj 2 ˛=n

kf kKP q˛;p .!1 ;!2 / :

Repeating the arguments used for the term F1 , we can also obtain F3  C KP q˛;p .!1 ; !2 /: Combining the above estimates for F1 ; F2 and F3 ; and taking the supremum for all  > 0; the proof of Theorem 2 is finished. 5. P ROOF OF T HEOREM 3 ˛;p Let f 2 KP q .!1 ; !2 /; then

Œb;  .f / p ˛;p ˝ KP q .!1 ;!2 / 0 1p jX 2 1 X ˛p




.Œb;  .f k /j .x/ q A !1 .Bj / n @ C ˝ L .!2 / jD 1

CC

1 X jD 1

kD 1

0
!1 .Bj /

˛p n

jX C1

@ kDj 1

1p

.Œb;  .f k /j .x/ q A ˝ L .!2 /

PARAMETRIC MARCINKIEWICZ INTEGRALS ON WEIGHTED HERZ SPACES

0 1p 1 X


˛p

.Œb;  .f k /j .x/ q A !1 .Bj / n @ ˝ L .!2 /

1 X

CC

881

jD 1

kDj C2

D G1 C G2 C G3 : 

By the fact that Œb; ˝  is a bounded operator on Lq .!2 /, we obtain 0 1p jX C1 1 X ˛p


p

f k q A  C kbkp G2  C !1 .Bj / n @  kf k P ˛;p L .!2 / jD 1

Kq

kDj 1

.!1 ;!2 /

:

Obviously,





Œb; ˝ .f k /j .x/

Lq .!2 /

Z C

Cj

Z CC

ˇq  bBk /˝ .fk /.x/ˇ !2 .x/dx

ˇ ˇ.b.x/

Cj

!1=q

ˇq bBk /fk /.x/ˇ !2 .x/dx

ˇ  ˇ ..b.
/ ˝

!1=q

D H1 C H2 : For the term G1 ; since j  k C 2; by (3.1) we have !1=q Z ˇ ˇq  ˇ.b.x/ bB / .fk /.x/ˇ !2 .x/dx H1 D C k ˝ Cj

 C2

jn

 C2

jn

!1=q

Z kf k kL1 kf k kL1

C .jbBj

Bj

jb.x/

 Z Bj

bBk jq !2 .x/dx

1=q bBj ;!2 jq !2 .x/dx Z
1=q  bBk j/ !2 .x/dx :

jb.x/

bBj ;!2 j C jbBj

Bj

BMO.Rn /,

By Lemma 4 and the definition of we have Z Z  1=q q jb.x/ bBj ;!2 j !2 .x/dx  C kbk . Bj

!2 .x/dx/1=q ;

(5.1)

Bj

jbBj

bBj ;!2 j  C kbk ;

(5.2)

and jbBj

bBk j  C.j

k/kbk :

(5.3)

882

YUE HU AND YUESHAN WANG

From (3.2),(3.3) and (5.1)-(5.3), we get H1  C kbk kf k kLq .!2 / .j  C kbk? kf k kLq .!2 / .j

jn

k/2

Z .

1=q

!2 .x/dx/

Z .

Bj

k/2.j

!2 .x/1

q0

dx/1=q

0

Bk

k/n.q2 =q 1/

for j  k C 2: Similar to the estimate of (3.1), ˇ  ˇ ˇ ..b.
/ bB /fk /.x/ˇ  C 2 k ˝

Z

jn

Ck

jb.´/

 bBk jjf .´/jd´ :

So Z Cj

 C2

ˇ  ˇ ..b.
/ ˝

jn

ˇq bBk /fk /.x/ˇ !2 .x/dx


1=q !2 .Bj /

Z Ck

jb.´/

!1=q

 bBk jjf .´/jd´ :

1 q0

Since !2 2 Aq2 ; by Lemma 1 we know that !2 2 2 Aq20 : Therefore by Lemma 4 we have Z 1=q 0 Z 0 0 q0 1 q0 jb.x/ bBk j !2 .x/ dx !2 .x/1 q dx/1=q : (5.4)  C kbk . Bk

Bk

Using H¨older’s inequality , (3.3) and (5.4) we get Z Z
1=q jn H2  C 2 jb.y/ bBk jjf k .y/jdy !2 .x/dx Bk

 C2

jn

Bj

Z

q0

jb.y/

1=q 0

1 q0

Z

bBk j !2 .y/ dy kf k kLq .!2 / !2 .x/dx Bk Bj Z Z
1=q 1 q0 jn 1=q 0  C kbk kf k kLq .!2 / 2 . !2 .x/ dx/ !2 .x/dx Bk

 C kbk kf k kLq .!2 / 2


1=q

Bj

.j k/n.q2 =q 1/

:

Summarizing the above estimates, we have that for j  k C 2;



  C kbk kf k kLq .!2 / .j k/2.j

Œb; ˝ .f k /j .x/ q L .!2 /

k/n.q2 =q 1/

:

(5.5) Using (5.5) and repeating the estimation process of E1 , we obtain p

p ˛;p KP q .!1 ;!2 /

G1  C kbk kf k for 0 < p < 1:

PARAMETRIC MARCINKIEWICZ INTEGRALS ON WEIGHTED HERZ SPACES

883

Finally, let us estimate G3 : Since j  k 2; by (3.4) we have !1=q Z ˇ ˇq  ˇ.b.x/ bB / .fk /.x/ˇ !2 .x/dx H1 D C k ˝ Cj

kn

 C2

!1=q

Z kf k kL1

jb.x/

Bj

q

bBk j !2 .x/dx

:

From (3.2),(5.1)-(5.3) and (3.6), we get H1  C kbk kf k kLq .!2 / .j  C kbk? kf k kLq .!2 / .j

jn

k/2

Z .

!2 .x/dx/

Z .

Bj

!2 .x/1

q0

dx/1=q

0

Bk

.j k/ı2 n=q

k/2

for j  k 2: Similar to the estimate of (3.4), ˇ  ˇ ˇ ..b.
/ bB /fk /.x/ˇ  C 2 ˝

1=q

kn

Z

k

Ck

jb.´/

 bBk jjf .´/jd´ :

So Z Cj

 C2

ˇq bBk /fk /.x/ˇ !2 .x/dx

ˇ  ˇ ..b.
/ ˝

kn


1=q !2 .Bj /

Z Ck

jb.´/

Using H¨older’s inequality, (5.4) and (3.6), Z Z kn H2  C 2 jb.y/ bBk jjf k .y/jdy  C2

 bBk jjf .´/jd´ :

!2 .x/dx


1=q

Bj

Bk

kn

!1=q

Z

q0

jb.y/

1 q0

1=q 0

Z

bBk j !2 .y/ dy !2 .x/dx kf k kLq .!2 / Bk Bj Z Z
1=q 1 q0 1=q 0 kn !2 .x/dx  C kbk kf k kLq .!2 / 2 . !2 .x/ dx/ Bj

Bk

.j k/ı2 n=q

 C kbk kf k kLq .!2 / 2

:

Thus



 Œb;  .f  / .x/

k j ˝

Lq .!2 /

for j  k

 C kbk kf k kLq .!2 / .k

j /2.j

2: Repeating the estimation process of E3 , we obtain p

G3  C kbk kf k for 0 < p < 1:

p ˛;p KP q .!1 ;!2 /

k/ı2 n=q


1=q

884

YUE HU AND YUESHAN WANG

Summing up the estimates of G1 ; G2 and G3 ; it completes the proof of Theorem 3. ACKNOWLEDGEMENT The author is very grateful to the anonymous referees and the editor for their insightful comments and suggestions. R EFERENCES [1] A. Al-Salman, “On the l 2 -boundedness of parametric marcinkiewicz integral operator,” J. Math. Anal. Appl., vol. 375, no. 2, pp. 745–752, 2011, doi: 10.1016/j.jmaa.2010.09.062. [2] S. Aliev and V. Guliev, “Boundedness of the parametric marcinkiewicz integral operator and its commutators on generalized morrey spaces,” Georgian Math. J., vol. 19, no. 2, pp. 195–208, 2012. [3] Q. Fang and X. Shi, “Estimates for parametric marcinkiewicz integrals in bmo and campanato spaces,” Appl. Math. J. Chinese Univ. Ser. B, vol. 26, no. 2, pp. 230–252, 2011, doi: 10.1007/s11766-011-2685-7. [4] L. Grafakos, “Classical and modern fourier analysis,” Pearson Education, Inc., Upper Saddle River, NJ, 2004. [5] Y. Guo and Y. Jiang, “Weighted herz space and regularity results,” J. Funct. Spaces Appl., no. 283730, p. 13, 2012. [6] L. H¨ormander, “Estimates for translation invariant operators in l p spaces,” Acta Math., vol. 104, pp. 93–140, 1960, doi: 10.1007/BF02547187. [7] F. John and L. Nirenberg, “On functions of bounded mean oscillation,” Comm. Pure Appl. Math., vol. 14, pp. 415–426, 1961, doi: 10.1002/cpa.3160140317. [8] Y. Komori and K. Matsuoka, “Boundedness of several operators on weighted herz spaces,” J. Funct. Spaces Appl., vol. 7, no. 1, pp. 1–12, 2009, doi: 10.1155/2009/739134. [9] S. Lu and D. Yang, “The decomposition of weighted herz space on Rn and its applications,” Sci. China Ser. A., vol. 38, no. 2, pp. 147–158, 1995. [10] B. Muckenhoupt, “Weighted norm inequalities for the hardy maximal function,” Trans. Amer. Math. Soc., vol. 165, pp. 207–226, 1972, doi: 10.1090/S0002-9947-1972-0293384-6. [11] B. Muckenhoupt and R. Wheeden, “Weighted bounded mean oscillation and the hilbert transform,” Studia Math., vol. 54, no. 3, pp. 221–237, 1975/76. [12] Z. F. S Gong and B. Ma, “Weighted multilinear hardy operators on herz type spaces,” The Scientific World Journal, no. 420408, p. 10, 2014. [13] K. Y. S. Lu and D. Yang, “Boundedness of some sublinear operators in weighted herz-type spaces,” Kodai Math. J., vol. 23, no. 3, pp. 391–410, 2000, doi: 10.2996/kmj/1138044267. [14] X. Shi and Y. Jiang, “Weighted boundedness of parametric marcinkiewicz integral and higher order commutator,” Anal. Theory Appl., vol. 25, no. 1, pp. 25–39, 2009, doi: 10.1007/s10496009-0025-z. [15] E. Stein, “On the functions of littlewood-paley,” Lusin and Marcinkiewicz, Trans. Amer. Math. Soc., vol. 88, pp. 430–466, 1958, doi: 10.1090/S0002-9947-1958-0112932-2. [16] A. Torchinsky and S. Wang, “A note on the marcinkiewicz integral,” Colloq. Math., vol. 60/61, no. 2, pp. 235–243, 1990. [17] H. Wang, “The boundedness of intrinsic square functions on the weighted herz spaces,” J. Funct. Spaces, no. 274521, p. 14, 2014. [18] Q. X. Y. Ding and K. Yabuta, “Ba remark to the l 2 boundedness of parametric marcinkiewicz integral,” J. Math. Anal. Appl., vol. 387, no. 2, pp. 691–697, 2012, doi: 10.1016/j.jmaa.2011.09.020.

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[19] Y. H. Y. Hu and Y. Wang, “The commutators of fractional integrals on generalized herz spaces,” J. Funct. Spaces, no. 428493, p. 6, 2014.

Authors’ addresses Yue Hu Henan Polytechnic University, College of Mathematics and Informatics, 454003, Jiaozuo, Henan, China E-mail address: [email protected] Yueshan Wang Jiaozuo University, Department of Mathematics, 454003, Jiaozuo, Henan, China E-mail address: [email protected]